It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ones (if one doesn't want to the limit with respect to all coverings, then my statement means that there exists a certain descent spectral sequence). This is certainly wrong for etale cohomology, since if we have a covering $U_1\sqcup U_2\to X$, we definitely need to consider the cohomology of $U_1\times_X U_1$ and of $U_2\times_X U_2$ (I don't know whether two copies of cohomology of $U_1\times_X U_2$ are really needed for the corresponding computation; yet this is not what I really want to ask). Now suppose that $X$ is (the spectrum of) a field (so, sheaves are just Galois representations, but still a certain (fixed) unordered Cech complex computes the cohomology of etale sheaves over $X$ (i.e. that we have the corresponding descent). Does this imply easily that the covering $\sqcup U_i\to X$ ($U_i$ are etale $X$-schemes, i.e. disjoint unions of spectra of finite separable field extensions of $X$) 'splits'? Were 'inverse' questions of this type considered anywhere already (usually one wants to prove that for a hypercovering or something like this there is descent; here I would like to use the fact that there is descent for something that usually is far from being a hypercover)?

Actually, I would like to understand a more general situation. The $n$-th level of 'my' complex is the cohomology of some pro-finite separable $X$-scheme that only maps into the '$n$-th ordered power' of $U$ over $X$, and the isomorphism of cohomology only holds in degrees $\le d$ (for a fixed $d>0$). Any hints or references here would also be very welcome!

P.S. It seems that in the second setting I get a split surjection from the 'unordered Cech' i.e. of 'the true' cohomology onto the the 'ordered' i.e. 'false' one. Probably, substituting sheaves (i.e. Galois representations) corresponding to the relevant Galois algebras, I will be able to prove a certain splitting result. Yet I would really like to know whether there are any 'conceptual' methods for dealing with this situation.

  • $\begingroup$ I don't understand what you mean by 'Does this imply easily that the covering ... 'splits'?' ? It seems to me that Deligne's theory of cohomological descent fits to what you're asking for in your first paragraph: see Brian Conrad's notes math.stanford.edu/~conrad/papers/hypercover.pdf (esp. Thm 7.2 can be interpreted to say that 'there is descent for something that usually is far from being a hypercover') and the references given there (Expose Vbis in SGA4, Deligne's Theorie des Hodge III) for details. $\endgroup$ Oct 18, 2011 at 16:05
  • $\begingroup$ Well, I have written that $X$ is a field; this is really a strong restriction. $\endgroup$ Oct 18, 2011 at 20:08


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